Two boxes contain exactly the same total number of balls, and in each box some balls are white and the rest are black. Let x and y be the number of white and black balls in the first box, respectively, and let z be the number of white balls in the second box. From each box, we pick up n balls with replacement. Find the conditions under which the following two quantities are equal:

(i) The probability all balls selected from the second ball are white.

(ii) The probability that the balls chosen from the first box are all of the same color, i.e. they are all white or all black.

Show that finding positive integers n, x, y, z such that the condition for the above two probabilities to be equal amounts to finding positive integers that satisfy the equation xn + yn = zn (the solution to this problem is very well-known4 as

“Fermat’s last theorem.”